Chaos, Fractionality, Nonlinear Contagion, and Causality Dynamics of the Metaverse, Energy Consumption, and Environmental Pollution: Markov-Switching Generalized Autoregressive Conditional Heteroskedasticity Copula and Causality Methods
Abstract
:1. Introduction
2. Methodology
3. Empirical Results
3.1. Data
3.2. Results
- Long-term dependence and fractionality. Fractionality and long-term dependence are examined for the dataset with Malderbrot and Wallis’ R/S and Lo’s modified R/S tests. Long-memory and fractional parameters are estimated, and further tests for statistical significance are conducted with the Geweke and Porter–Hudak [24] and Robinson–Henry methods [25].
- Determination of chaos, entropy, and complexity. To determine chaotic dynamics, the largest Lyapunov (λ) [21] exponents are calculated. The λ results provide insight into the average exponential rate of divergence or convergence of nearby orbits in the phase space. A positive Lyapunov exponent (λ > 0) indicates chaos, signifying the divergence of nearby trajectories. Conversely, when λ ≤ 0, the time series corresponds to regular motion. A λ value equal to zero indicates a bifurcation. A λ value greater than 1 signifies a deterministic chaotic process. If λ ≤ 0 and λ is greater than 1, we do not explore cointegration and Granger causality tests. However, when λ is less than 1 and falls between 0 and 1 (0 < λ < 1) and the Shannon entropy yields a positive value, it suggests uncertainty or a random process. Entropy is evaluated with Shannon entropy [22] and by Havrda–Charvât–Tsallis (HCT) entropy measures [32,33]. Kolmogorov–Sinai complexity (KS) is evaluated following the method of [34]. The measures above not only give information on the existence of chaos but also the degree of randomness and complexity. Following the steps above, we delve into the nonlinearities and the stochastic process of the variables.
- Unit root and stationarity testing. Following BDS tests, the unit root and stationary are tested with linear and nonlinear methods. To this end, traditional linear Augmented Dickey–Fuller (ADF) [54] is followed by a nonlinear Kapetanios et al. (KSS) [55] unit root, and the Kwiatowski et al. (KPSS) [56] test of stationarity is used. While the first assumes a linear unit root, the second tests the unit root against smooth transition autoregressive (STAR) type nonlinearity. Lastly, KPSS is a nonparametric test known to perform well under series with structural breaks and nonlinearity.
- Modeling series and their relationships are modeled for their marginal and joint distributions with Markov-Switching Generalized Autoregressive Conditional Heteroskedasticity Copula (MS-GARCH–Copula) modeling. The method allows for the modeling of nonlinear and heteroskedastic series for their tail dependence and contagion dynamics.
- Causality testing. The MS-GARCH–Copula Causality test is an extension of linear Granger causality testing to nonlinear and heteroskedastic series. The method benefits from GARCH–Copula and causality tests.
- Comparative analysis and robustness. After evaluating the models with diagnostics tests and following nonlinear causality testing in the previous stage, the directions of causality are determined under distinct regimes. At this stage, the results are compared to single-regime GARCH–Copula Causality test results. The findings are examined for possible deviations and differences from the nonlinear methods.
3.2.1. Long-Term Dependence and Fractionality
3.2.2. Chaos and Entropy Tests
3.2.3. Nonlinearity Test Results
3.2.4. MS-GARCH–Copula Estimation Results
3.2.5. Nonlinear Causality Test Results with Regime-Switching and Copulae
3.3. Discussion with Comparisons to the Existent Literature
4. Conclusions and Policy Recommendations
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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CO2 | EC | MV | |
---|---|---|---|
Mean | 0.000016 | 0.000277 | 0.000222 |
Med. | 0.000025 | 0.000436 | 0.000333 |
Mx. | 0.010198 | 0.127332 | 0.021183 |
Mn. | −0.01024 | −0.115797 | −0.028083 |
Skw. | −0.613858 | 0.036475 | −0.526506 |
Kr. | 6.750716 | 20.54980 | 8.171633 |
JB | 2628.4308 [0.0000] | 33495.09 [0.0000] | 3029.190 [0.0000] |
Data explanation and sources | Daily CO2 emission concentration, given in PPM, taken as a proxy of air pollution. Source: CO2 Earth Database, Publicly available at https://www.co2.earth/ (accessed on 1 January 2024) | Energy consumption of Bitcoin, in Twh. Source: Uni. of Cambridge, Center for Alternative Finance. Publicly available at https://ccaf.io/cbnsi/cbeci (accessed on 1 January 2024) | Extended version of Roundhill’s metaverse index/Source: Authors’ own calculations following Roundhill’s method. Source: Refinitiv Eikon Database. |
Tests/Series | CO2 | EC | MV |
---|---|---|---|
Series in levels | |||
Mandelbrot–Wallis R/S test statistic | 22.3131 *** | 21.4118 *** | 20.7831 *** |
Lo’s modified R/S test statistic | 15.7823 *** | 15.1506 *** | 14.7048 *** |
d-parameter (s.e.) [p-value] | 0.990215 *** (0.01870) [0.0000] | 0.99511 *** (0.01901) [0.0000] | 0.994037 *** (0.01869) [0.0000] |
Series, first differenced | |||
Mandelbrot–Wallis R/S test statistic | 3.06288 *** | 0.768264 | 1.52558 |
Lo’s modified R/S test statistic | 2.17338 *** | 0.80196 | 1.56507 |
d-parameter (s.e.) [p-value] | 0.0525454 (0.175986) [0.7653] | −0.0395671 ** (0.018712) [0.0345] | −0.0193269 (0.018706) [0.3015] |
Volatility series | |||
Mandelbrot–Wallis R/S test statistic | 3.80497 *** | 3.90547 *** | 4.31435 *** |
Lo’s modified R/S test statistic | 5.35633 *** | 3.20997 *** | 3.77577 *** |
d-parameter (s.e.) [p-value] | 0.84098 *** (0.01401) [0.0000] | 0.475363 *** (0.013012) [0.0000] | 0.263036 *** (0.013857) [0.0000] |
Dimension | CO2 | MV | EC |
---|---|---|---|
2 | 0.113 | 0.568 | 0.767 |
3 | 0.135 | 0.675 | 0.814 |
4 | 0.129 | 0.632 | 0.855 |
5 | 0.111 | 0.599 | 0.876 |
6 | 0.112 | 0.610 | 0.887 |
CO2 | EC | MV | |
---|---|---|---|
Shannon entropy | 7.392 | 10.589 | 11.238 |
Shannon, transformed to the [0, 1] range 1 | 0.513 | 0.464 | 0.472 |
Havrda–Charvât–Tsallis (HCT) measure | 55.612 | 54.409 | 52.501 |
Kolmogorov–Sinai (KS) complexity measure | 8.145 | 10.589 | 11.238 |
Dimension | BDS Statistic | Std. Error | z-Statistic | Prob. |
---|---|---|---|---|
MV | ||||
2 | 0.016624 | 0.001766 | 9.414265 | 0.0000 |
3 | 0.035795 | 0.002800 | 12.78438 | 0.0000 |
4 | 0.050123 | 0.003327 | 15.06666 | 0.0000 |
5 | 0.059543 | 0.003460 | 17.20966 | 0.0000 |
6 | 0.063227 | 0.003329 | 18.99010 | 0.0000 |
CO2 | ||||
2 | 0.030968 | 0.002296 | 13.48518 | 0.0000 |
3 | 0.052432 | 0.003657 | 14.33557 | 0.0000 |
4 | 0.064923 | 0.004367 | 14.86660 | 0.0000 |
5 | 0.076177 | 0.004565 | 16.68686 | 0.0000 |
6 | 0.082396 | 0.004416 | 18.65772 | 0.0000 |
EC | ||||
2 | 0.186349 | 0.001552 | 120.0759 | 0.0000 |
3 | 0.322729 | 0.002453 | 131.5776 | 0.0000 |
4 | 0.418268 | 0.002904 | 144.0096 | 0.0000 |
5 | 0.483784 | 0.003010 | 160.7141 | 0.0000 |
6 | 0.528045 | 0.002887 | 182.9322 | 0.0000 |
MV | CO2 | EC | |
---|---|---|---|
ARCH-LM(1–5) | 2287.181 | 223.3280 | 625.9534 |
Result: | ARCH effects | ARCH effects | ARCH effects |
White LM test | 1102.55 | 550.64 | 1112.81 |
Result: | Heteroskedasticity | Heteroskedasticity | Heteroskedasticity |
ADF | −51.45 | −4.306 | −9.739 |
KSS | −8.37 | −85.95 | −23.29 |
KPSS | 0.091 | 0.019 | 0.332 |
Results: | I(1) | I(1) | I(1) |
ARCH | GARCH | Cons. | P(1|1) | P(2|2) | Model Fit and Diagnostic Tests | |
---|---|---|---|---|---|---|
Dependent Variable: Metaverse | ||||||
Regime 1 | 0.177220 *** (2.69) 1 | 0.699015 *** (4.74) | 0.1231 ** (3.44) | 0.768402 *** | 0.88031 *** | LogL: 2480.11 RMSE:1.002347, MAE: 1.002345, ARCH-LM: 0.0395 |
Regime 2 | 0.1134 *** (3.47) | 0.8142 * (2.59) | 0.0727 ** (2.23) | |||
Dependent Variable: Energy Consumption of Bitcoin | ||||||
Regime 1 | 0.2417 ** (2.83) | 0.682 *** (2.76) | 0.077 ** (2.44) | 0.715692 *** | 0.83704 *** | LogL: 18117, RMSE:0.0118, MAE:0.0078, ARCH-LM: 0.026 |
Regime 2 | 0.189579 (2.91) | 0.731939 (3.07) | 0.0796 ** (2.66) | |||
Dependent Variable: CO2 Emission Concentration | ||||||
Regime 1 | 0.40027 ** (2.09) | 0.58881 *** (3.06) | −0.000156 * (1.77) | 0.71011 *** | 0.86001 *** | LogL: 2287.44, RMSE: 0.132, MAE: 0.118, ARCH-LM: 0.15 |
Regime 2 | 0.4014 ** (2.45) | 0.590944 *** (3.15) | −0.0112 ** (2.465) | |||
Copula Results | ||||||
Regime 1 | Regime 2 | |||||
CO2-EC | CO2-MV | EC-MV | CO2-EC | CO2-MV | EC-MV | |
0.527 *** | 0.7991 *** | 0.569 *** | 0.5665 *** | 0.7614 *** | 0.61 *** |
Direction of Causality under H1: | Regime 1 | Causality? | Result: | Regime 2 | Causality? | Result: |
---|---|---|---|---|---|---|
CO2→MV MV→CO2 | 1.00102 1 | No | MV→CO2 (Unidirect.) | 0.87728 | No | MV→CO2 (Unidirect.) |
3.31528 ** | Yes | 3.62045 ** | Yes | |||
EC→CO2 CO2→EC | 0.17533 | No | CO2→EC (Unidirect.) | 1.22553 | No | CO2→EC (Unidirect.) |
2.74820 ** | Yes | 2.77701 ** | Yes | |||
MV→EC EC→MV | 3.04436 ** | Yes | MV→EC (Unidirect.) | 2.94021 ** | Yes | MV⇄EC (Bidirect.) |
1.40628 | No | 2.98736 ** | Yes |
Direction of Causality under H1: | Test Statistic 1 | Causality? | Result: |
---|---|---|---|
CO2→MV MV→CO2 | 0.51875 | No | MV→CO2 (Unidirectional) |
2.51875 * | Yes | ||
CO2→EC EC→CO2 | 1.11234 | No | No causality |
0.74383 | No | ||
EC→MV MV→EC | 8.63478 ** | Yes | MV⇄EC (Bidirectional) |
9.91133 ** | Yes |
Direction of Causality: | MS-ARMA-GARCH Copula Causality | ARMA-GARCH-Copula Causality | Difference in Results? | |
---|---|---|---|---|
Regime 1 | Regime 2 | Single Regime | ||
CO2→MV | MV→CO2 (Unidirectional) | MV→CO2 (Unidirectional) | MV→CO2 | No |
MV→CO2 | ||||
CO2→EC | CO2→EP (Unidirectional) | CO2→EP (Unidirectional) | No causality between CO2 and EP | Yes |
EC→CO2 | ||||
EC→MV | MV→EP (Unidirectional) | MV⇄EP (Bidirectional) | MV⇄EP (Bidirectional) | Yes, but similar to Regime 2 |
MV→EC |
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Bildirici, M.; Ersin, Ö.Ö.; Ibrahim, B. Chaos, Fractionality, Nonlinear Contagion, and Causality Dynamics of the Metaverse, Energy Consumption, and Environmental Pollution: Markov-Switching Generalized Autoregressive Conditional Heteroskedasticity Copula and Causality Methods. Fractal Fract. 2024, 8, 114. https://doi.org/10.3390/fractalfract8020114
Bildirici M, Ersin ÖÖ, Ibrahim B. Chaos, Fractionality, Nonlinear Contagion, and Causality Dynamics of the Metaverse, Energy Consumption, and Environmental Pollution: Markov-Switching Generalized Autoregressive Conditional Heteroskedasticity Copula and Causality Methods. Fractal and Fractional. 2024; 8(2):114. https://doi.org/10.3390/fractalfract8020114
Chicago/Turabian StyleBildirici, Melike, Özgür Ömer Ersin, and Blend Ibrahim. 2024. "Chaos, Fractionality, Nonlinear Contagion, and Causality Dynamics of the Metaverse, Energy Consumption, and Environmental Pollution: Markov-Switching Generalized Autoregressive Conditional Heteroskedasticity Copula and Causality Methods" Fractal and Fractional 8, no. 2: 114. https://doi.org/10.3390/fractalfract8020114
APA StyleBildirici, M., Ersin, Ö. Ö., & Ibrahim, B. (2024). Chaos, Fractionality, Nonlinear Contagion, and Causality Dynamics of the Metaverse, Energy Consumption, and Environmental Pollution: Markov-Switching Generalized Autoregressive Conditional Heteroskedasticity Copula and Causality Methods. Fractal and Fractional, 8(2), 114. https://doi.org/10.3390/fractalfract8020114